prove that equation has only 1 solution

Question

prove that the equation : $4ax^3+3bx^2+2cx=a+b+c$ have only one solution.

what I did so far was to define $g(x)=ax^4+bx^3+cx^2-(a+b+c)x$ so that $g(0)=0$ and also $g(1)=a+b+c-(a+b+c)=0$ so by Rolle's theorem exist a point $d$ such that $g'(d)=0=4ad^3+3bd^2+2cd-a+b+c$. so $4ad^3+3bd^2+2cd=a+b+c$ but I have no idea how to prove that there isn't another solution.

Answer 1

It's wrong.

Try $a=c=0$ and $b=1$.